T-185: A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition

V3 C3 Upper bound confirmed superseded by T-186

2026-10-11 published · Couzo after Xu, Daniel, Ellsworth, Levy · $n = 175$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (175) \leq 13.7671190241983642888907662905$ . The selected standing case is unchanged.

The three retained jobs at 175, the certificate with its duplicate-square and outside-container controls, were decided again here on 16 October 2026 by both maintained exact routes and equal their retained rows, and the frozen comparison rebuilt equal. One maintained route is the verifier the source reports running; the independent rational corner checker is independently re-implemented, and the replay rests on it. The 16 October review found no blocking defect.

It is the release's certificate at 175, first committed on 11 October 2026, which issue 506's release holds and the issue does not name; the author asked for it to be imported on issue 506 on 14 October 2026. When it was published it was below the case's ceiling, Xu's packing (T-125), by 1.78e-03, and below every other report at 175: T-140 (issue 488) by 1.70e-06, T-130, T-138 and the issue-476 certificate by 3.61e-05, and T-101 (issue 375) by 1.11e-02. Ostashev's issue 528 composition (T-158), the case's house since 12 October 2026, is 2.94e-03 below it, so the entry is superseded.

Credit Couzo (https://github.com/franciscouzo/square-packing), after Xu: the source's standing table says the 175 was seeded from Xu. Measured here, it is the arrangement of his T-140 certificate, which started from Xu's packing, refined: every centre within 0.0113 of T-140's and 137 of 175 within 1e-3. By pose no symmetry pairs it, or T-140, with Xu's packing (T-125) one to one, at most 86 of its 175 squares lying within a tenth of Xu's (the 16 October review, RT-2). Issue 506 states that the search code and the request were written with Claude (Claude Code), directed and reviewed by the author.

Significance, composition and next rung

Significance: A refinement of the author's own T-140 packing at $n = 175$ to a new smallest side when published, 1.70e-06 below it and 1.78e-03 below the case ceiling then, as T-165 and T-167 are; not padding; no lower bound or optimum.
Next rung: None for the bound, which T-158 supersedes at 175. V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 16 October, is retained.
Regularizations: n = 175: a regularization of T-140’s packing by Couzo (exact optimum from Daniel's exact contact solver, seeded from Xu; solved to its side; identity: measured here; 38 squares moved, at most 0.01122).
Novelty: previously-published Present in an identified source

The case

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, Ostashev 2026 [Mishapolk KKT refinements 2026-10-14]

13.2313.764

131415

13.22914.229

\sqrt{n}

\sqrt{n} + 1

Proven

$13.23773$ $\leq s (175) \leq$ $13.764175$

exact

Citation record n-175

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperOstashev et al., GitHub (confirmed T-186)

Open

optimality

The case record

LowerUpper

Verified13.23773920…13.76417431…

Reported13.30130208…13.76417431…

Gap0.52643511…

Results on the case

14 results in the register on $n = 175$ , oldest first, each with what it established and how it stands now.

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-101
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
V3 C3 Upper bound confirmed on this case, superseded by T-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-08 published T-125
Complete rational construction reports at 25 counts
V3 C3 Upper bound confirmed superseded by T-186
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-130
Five follow-up rational refinements from Couzo
V3 C3 Upper bound confirmed superseded by T-186
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-138
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
V3 C3 Upper bound confirmed superseded by T-186
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-140
Six more exact rational certificates from Couzo
V3 C3 Upper bound confirmed superseded by T-186
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-11 published T-185 this result
A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition
V3 C3 Upper bound confirmed superseded by T-186
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register
2026-10-12 published T-158
New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly
V3 C3 Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register
2026-10-13 published T-174
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
V3 C3 Upper bound confirmed on this case, superseded by T-186
PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register
2026-10-13 published T-180
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
V0 C1 Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-14 published T-186
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-16 published T-193
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
V3 C3 Upper bound confirmed on this case, superseded by T-186
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

Links

On this site: Case record, $n = 175$ · Frontier row, $n = 175$ · T-185 in the results table

On GitHub, at main

Register: T-185 in results.yaml, line 25237
Evidence: E-couzo-506-n175-report · E-couzo-506-n175-exact-feasibility
Proofs and certificates: certificate certificates
Sources: Couzo n175 2026-10-11
Source packet: resources/web/couzo-certificates-2026-10-11/README.md
Artifacts: 6 artifacts and controls
resources/web/couzo-certificates-2026-10-11/README.md resources/web/couzo-certificates-2026-10-11/acquisition/claims.json resources/web/couzo-certificates-2026-10-11/source/certificates/n175.cert devtools/upper_bound_reports.py docs/project/reviews/review-2026-10-16-upper-bound-imports-506-607.md tests/test_upper_bound_reports.py
Case file: frontier/n-175.md (verified lower, verified upper, reported lower, reported upper)